Differences Between Robin and Neumann Eigenvalues

Let $$ be a bounded planar domain, with piecewise smooth boundary $$ . For $$ , we consider the Robin boundary value problem where $$ is the derivative in the direction of the outward pointing normal to $$ . Let $$ be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps For a wide class of planar domains we show that there is a limiting mean value, equal to $$ and in the smooth case, give an upper bound of $$ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.